There are so many factors that play into this, most have been covered. This is what I weigh:
Do I really care about what they are demanding?
No. Reasons to give in - Take it and go away. I don't have time for your childish games.
Reasons to say no - I'm looking to pick a fight anyway. Volunteers are always welcome, particularly when they come with the gift of war happiness. Look at all the friends I invited to the party! Besides, when the AI goes to war, research usually suffers on their end. Its a good way to slow them down (less investment and fewer trading partners).
Yes. Reasons to give in I dread what will happen if I don't. I don't want to spend the resources to fight the AI. I don't want to halt my expansion to fight over the last pretzel in the bowl. I don't want to end up at the bottom of a dogpile. All alternatives appear more expensive.
Reasons to say no - You can pry it from my cold, dead hands. Even if you were the last Civ on earth I wouldnt give you the satisfaction.
Can they reach me (or reach me in a reasonable amount of time)?
No. Reasons to give in - no chance of war, keep trading partners for techs/luxuries.
Reasons to say no - the AI will never suicide galley, let them stew on the other side of the world and enjoy the war happiness. Danger - they get other AI's into the war or ones that are closer to you or you end up unable to trade techs effectively/efficiently.
Yes. Reasons to give in - don't want to fight a pointless war over 10gp and a map. I'd rather build then conquer, Keep trading partners.
Reasons to say no - the AI may declare. Gain war happiness and fish for leaders as units straggle into my territory. Its time to expand and they just volunteered for a national realignment of their interests with mine.
Are they stronger than me?
No. Reason to give in - basically none, but might not want to bother with fighting an enemy I can't reach or risk the AI starting a world war because I wouldn't give him my left shoe. World wars limit trading options and money making opportunities.
Reason to say no - I'm ready to claim their land in the name of the one true and lawful nation - MINE. Also, if they declare on me I'm in no real danger - enjoy war happiness and fish for leaders.
Yes. Reasons to give in - Don't want to fight a war of attrition against a stronger enemy or pay even more to bribe other AIs into helping beat up on the bigger kid. Not ready to take them on and have no real hope of gaining any territory/resources from a clash. Possible dogpile with associated problems.
Reasons to say no - The AI is getting out of control and it is time to take them down a notch. Start a dogpile on them and isolate them from trading partners. I'm ready to leader fish and the war happiness is always nice. They are asking for something I just won't give up - like that shiny new tech that I am trading around but they can't afford.
Do I have the resources to pull in allies to help in the fight?
No. Reasons to give in - the AI is stronger and has the means to start a dogpile on me. Aside from the possible military problems, this will also isolate me from trading techs and other items.
Reasons to say no - I am strong enough to stand alone and I don't care if a dogpile starts; I'm the tech leader and a military powerhouse that can fend off each spearman as they individually land on my coast. Note: avoid using tanks, just in case.
Yes. Reasons to give in - the other AI is stronger and it will cost me more to bring in my allies than to just pay them off. The AI is strong enough to bride some allies of its own and world wars limit trading possibilities.
Reasons to say no - Let them declare and start a dogpile. Receive war happiness while isolating the aggressive AI from the trade network.
What I dont ask myself:
Will this make me appear weak? The AI doesnt care how many times you buckled to demands. This is about relative strength the AI will make a demand based on aggression and if it calculates it has the strength to take it.
Can my pride handle this? Or- They cant do that to me! Hmmm. . . . Pride doesnt matter. Ill even the score later; either by war or making the AI pay through the nose for a tech, luxury or resource.